Properties

Label 3.4.2.2
Base \(\Q_{3}\)
Degree \(4\)
e \(2\)
f \(2\)
c \(2\)
Galois group $C_4$ (as 4T1)

Related objects

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Defining polynomial

\(x^{4} - 3 x^{2} + 18\)  Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $4$
Ramification exponent $e$: $2$
Residue field degree $f$: $2$
Discriminant exponent $c$: $2$
Discriminant root field: $\Q_{3}(\sqrt{2})$
Root number: $-1$
$|\Gal(K/\Q_{ 3 })|$: $4$
This field is Galois and abelian over $\Q_{3}.$

Intermediate fields

$\Q_{3}(\sqrt{2})$

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{3}(\sqrt{2})$ $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{2} - x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{2} - 3 t \)$\ \in\Q_{3}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_4$ (as 4T1)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$2$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{4} - x^{3} - 4 x^{2} + 4 x + 1$